1. 1
2. 2
3. 3
4. 4
5. 5
6. 6
7. 7
8. 8
9. 9
10. 10
11. 11
12. 12
13. 13
14. 14
15. 15
16. 16
17. 17
18. 18
19. 19
20. 20
21. 21
22. 22
23. 23
24. 24
25. 25
26. 26
27. 27
28. 28
29. 29
30. 30
31. 31
32. 32
33. 33
34. 34
35. 35
36. 36
37. 37
38. 38
39. 39
40. 40
41. 41
42. 42
43. 43
44. 44
45. 45
46. 46
47. 47
48. 48
49. 49
50. 50
51. 51
52. 52
53. 53
54. 54
55. 55
56. 56
57. 57
58. 58
59. 59
60. 60
61. 61
62. 62
63. 63
64. 64
65. 65
66. 66
67. 67
68. 68
69. 69
70. 70
71. 71
72. 72
73. 73
74. 74
75. 75
76. 76
77. 77
78. 78
79. 79
80. 80
81. 81
82. 82
83. 83
84. 84
85. 85
86. 86
87. 87
88. 88
89. 89
90. 90
91. 91
92. 92
93. 93
94. 94
95. 95
96. 96
97. 97
98. 98
99. 99
100. 100
101. 101
102. 102
103. 103
104. 104
105. 105
106. 106
107. 107
108. 108
109. 109
110. 110
111. 111
112. 112
113. 113
114. 114
115. 115
116. 116
117. 117
118. 118
119. 119
120. 120
121. 121
122. 122
123. 123
124. 124
125. 125
126. 126
127. 127
128. 128
129. 129
130. 130
131. 131
132. 132
133. 133
134. 134
135. 135
136. 136
137. 137
138. 138
139. 139
140. 140
141. 141
142. 142
143. 143
144. 144
145. 145
146. 146
147. 147
148. 148
149. 149
150. 150
151. 151
152. 152
153. 153
154. 154
155. 155
156. 156
157. 157
158. 158
159. 159
160. 160
161. 161
162. 162
163. 163
164. 164
165. 165
166. 166
167. 167
168. 168
169. 169
170. 170
171. 171
172. 172
173. 173
174. 174
175. 175
176. 176
177. 177
178. 178
179. 179
180. 180
181. 181
182. 182
183. 183
184. 184
185. 185
186. 186
187. 187
188. 188
189. 189
190. 190
191. 191
192. 192
193. 193
194. 194
195. 195
196. 196
197. 197
198. 198
199. 199
200. 200
201. 201
202. 202
203. 203
204. 204
205. 205
206. 206
207. 207
208. 208
209. 209
210. 210
211. 211
212. 212
213. 213
214. 214
215. 215
216. 216

import data.set.lattice
import data.set.function
import analysis.special_functions.log.basic

section

variables {α β : Type*}
variable  f : α → β
variables s t : set α
variables u v : set β
open function
open set

example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v :=
by { ext, refl }

example : f '' (s ∪ t) = f '' s ∪ f '' t :=
begin
  ext y, split,
  { rintros ⟨x, xs | xt, rfl⟩,
    { left, use [x, xs] },
    right, use [x, xt] },
  rintros (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩),
  { use [x, or.inl xs] },
  use [x, or.inr xt]
end

example : s ⊆ f ⁻¹' (f '' s) :=
begin
  intros x xs,
  show f x ∈ f '' s,
  use [x, xs]
end

example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v :=
sorry

example (h : injective f) : f ⁻¹' (f '' s) ⊆ s :=
sorry

example : f '' (f⁻¹' u) ⊆ u :=
sorry

example (h : surjective f) : u ⊆ f '' (f⁻¹' u) :=
sorry

example (h : s ⊆ t) : f '' s ⊆ f '' t :=
sorry

example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v :=
sorry

example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v :=
sorry

example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t :=
sorry

example (h : injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) :=
sorry

example : f '' s \ f '' t ⊆ f '' (s \ t) :=
sorry

example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) :=
sorry

example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) :=
sorry

example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u :=
sorry

example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) :=
sorry

example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) :=
sorry

variables {I : Type*} (A : I → set α) (B : I → set β)

example : f '' (⋃ i, A i) = ⋃ i, f '' A i :=
begin
  ext y, simp,
  split,
  { rintros ⟨x, ⟨i, xAi⟩, fxeq⟩,
    use [i, x, xAi, fxeq] },
  rintros ⟨i, x, xAi, fxeq⟩,
  exact ⟨x, ⟨i, xAi⟩, fxeq⟩
end

example : f '' (⋂ i, A i) ⊆ ⋂ i, f '' A i :=
begin
  intro y, simp,
  intros x h fxeq i,
  use [x, h i, fxeq],
end

example (i : I) (injf : injective f) :
  (⋂ i, f '' A i) ⊆ f '' (⋂ i, A i) :=
begin
  intro y, simp,
  intro h,
  rcases h i with ⟨x, xAi, fxeq⟩,
  use x, split,
  { intro i',
    rcases h i' with ⟨x', x'Ai, fx'eq⟩,
    have : f x = f x', by rw [fxeq, fx'eq],
    have : x = x', from injf this,
    rw this,
    exact x'Ai },
  exact fxeq
end

example : f ⁻¹' (⋃ i, B i) = ⋃ i, f ⁻¹' (B i) :=
by { ext x, simp }

example : f ⁻¹' (⋂ i, B i) = ⋂ i, f ⁻¹' (B i) :=
by { ext x, simp }

end

section
open set real

example : inj_on log { x | x > 0 } :=
begin
  intros x xpos y ypos,
  intro e,   -- log x = log y
  calc
    x   = exp (log x) : by rw exp_log xpos
    ... = exp (log y) : by rw e
    ... = y           : by rw exp_log ypos
end

example : range exp = { y | y > 0 } :=
begin
  ext y, split,
  { rintros ⟨x, rfl⟩,
    apply exp_pos },
  intro ypos,
  use log y,
  rw exp_log ypos
end

example : inj_on sqrt { x | x ≥ 0 } :=
sorry

example : inj_on (λ x, x^2) { x : ℝ | x ≥ 0 } :=
sorry

example : sqrt '' { x | x ≥ 0 } = {y | y ≥ 0} :=
sorry

example : range (λ x, x^2) = {y : ℝ  | y ≥ 0} :=
sorry

end

section
variables {α β : Type*} [inhabited α]

#check (default : α)

variables (P : α → Prop) (h : ∃ x, P x)

#check classical.some h

example : P (classical.some h) := classical.some_spec h

noncomputable theory
open_locale classical

def inverse (f : α → β) : β → α :=
λ y : β, if h : ∃ x, f x = y then classical.some h else default

theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) :
  f (inverse f y) = y :=
begin
  rw inverse, dsimp, rw dif_pos h,
  exact classical.some_spec h
end

variable  f : α → β
open function

example : injective f ↔ left_inverse (inverse f) f  :=
sorry

example : surjective f ↔ right_inverse (inverse f) f :=
sorry

end

section
variable {α : Type*}
open function

theorem Cantor : ∀ f : α → set α, ¬ surjective f :=
begin
  intros f surjf,
  let S := { i | i ∉ f i},
  rcases surjf S with ⟨j, h⟩,
  have h₁ : j ∉ f j,
  { intro h',
    have : j ∉ f j,
      { by rwa h at h' },
    contradiction },
  have h₂ : j ∈ S,
    sorry,
  have h₃ : j ∉ S,
    sorry,
  contradiction
end

end